The first effect which was discussed in detail in Ref. [1]} is due to the increase in the effective number of degrees of freedom.
Through the transition, as the energy density increases, the expansion rate of the universe increases accordingly.
As a result, there is an \(\ell \)  dependent phase-shift of the CMB power spectrum compared to a model of interacting radiation without a step.
To leading order there is no phase shift for small \(k\) -modes that enter the horizon after the transition,
while there is a linear phase-shift for large \(k\) -modes that enter the horizon before the transition.
Therefore, as the change in the behavior of the phase shift depends on the time of transition,
the CMB is sensitive to \(m_\phi \)  or equivalently to \(z_t\) .
Ref. [1]} found that this shift is strongly preferred by the full data set including measurements of \(H_0\)  and allows for better fits with higher expansion rates.
